which REST API is available for invoking using curl "force offline of node members" and checking the "current connections" for the node member.

If you are after calls equivalent to 'tmsh modify ltm <node> state user-down' and 'tmsh show sys connection", they are:

curl -sku <user>:<pass> https://<host>/mgmt/tm/ltm/node/<node> -X PATCH -H 'Content-type: application/json' -d '{"state":"user-down"}'

curl -sku <user>:<pass> https://<host>/mgmt/tm/sys/connection

Note that '/mgmt/tm/sys/connection" may return a large data, which may clog the management port. You may need to filter. For example, to filter by a client-side client address ('tmsh show sys connection cs-client-addr <addr>'),

curl -sku <user>:<pass> https://<host>/mgmt/tm/sys/connection?options=cs-client-addr,<addr>